Finite Group Databasebeta

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#72265618

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

No aliases for this group.

Subgroups

3953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 40 points:

(3,7)(5,8)(10,15,17,28)(12,22)(13,25)(16,27,29,38)(18,31,26,23)(20,34)(21,33)(24,37)(30,39,36,35)(32,40);(1,2)(3,8)(4,6)(5,7)(9,10,13,26,14,17,25,18)(11,16,21,36,19,29,33,30)(12,23,22,15,24,31,37,28)(20,35,34,27,32,39,40,38);(9,11)(10,16)(12,20)(13,21)(14,19)(15,27)(17,29)(18,30)(22,34)(23,35)(24,32)(25,33)(26,36)(28,38)(31,39)(37,40);(1,3,4,7)(2,5,6,8)(11,19)(12,24)(15,28)(16,29)(21,33)(22,37)(23,31)(30,36);(2,6)(5,8)

Polycyclic presentation:

2^10|a^2=g,b^2=gi,d^2=h,f^2=i,i^2=j,b^a=bf,d^a=dgh,d^b=dh,d^c=dj,e^b=eh,f^a=fi,f^b=fi,f^d=fj,g^b=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.7*F.9);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);